RN means that the individual has achieved the academic, clinical and legal requirements to be registered by a state board to practice as a nurse. It is a protected title, which means that only people so registered may use it. The actual academic, clinical and legal requirements vary from state to state and country to country - an RN may have a bachelor degree or (if older) passed through a hospital program.

Just wanted to add that those initials after names are called Credentials, whether it be college degrees or certifications. For example, a person who is an RN might also have earned a degree in nursing such as a Masters: thus the initials after that person's name would be RN, MSN. Say they have certified in a specialty like critical care through the appropriate organization, in this case American Association of Critical-Care Nurses (AACN). The credential they might earn through courses and reading and testing would be CCRN, which they can then use after their name.